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    $\begingroup$ Good, this is progress, showing that we cannot omit CH. But when CH holds, so far you've got only a disjoint family of size $\omega_1$. By my construction, one can push this to $\omega_2$, if there is a Kurepa tree, and to $2^{\omega_1}$, if there is a thick Kurepa tree. But do we really need these Kurepa trees? $\endgroup$
    – Joel David Hamkins
    Commented Mar 20, 2015 at 17:37
  • $\begingroup$ In my previous comment, I meant: almost-disjoint $\endgroup$
    – Joel David Hamkins
    Commented Mar 20, 2015 at 17:51
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    $\begingroup$ CH wouldn't be sufficient to get $\omega_2$ many almost disjoint selectors. For example, if there is an $\omega_2$-saturated sigma ideal over $\omega_1$. $\endgroup$
    – Ashutosh
    Commented Mar 21, 2015 at 1:56
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    $\begingroup$ @Ashutosh Would it be possible for you to expand on your comment and post an answer? $\endgroup$
    – Joel David Hamkins
    Commented Mar 21, 2015 at 11:42